Uniqueness of K-polystable degenerations of Fano varieties

TL;DR

This paper proves the uniqueness of K-polystable degenerations of Q-Fano varieties and establishes the separatedness of their moduli stack, leading to a better understanding of their classification and automorphism groups.

Contribution

It demonstrates the uniqueness of K-polystable degenerations and shows the moduli stack of K-stable Q-Fano varieties is separated, advancing the moduli theory of Fano varieties.

Findings

K-polystable degenerations are unique

The moduli stack of K-stable Q-Fano varieties is separated

Automorphism groups of K-stable Q-Fano varieties are finite

Abstract

We prove that K-polystable degenerations of Q-Fano varieties are unique. Furthermore, we show that the moduli stack of K-stable Q-Fano varieties is separated. Together with [Jia17,BL18], the latter result yields a separated Deligne-Mumford stack parametrizing all uniformly K-stable Q-Fano varieties of fixed dimension and volume. The result also implies that the automorphism group of a K-stable Q-Fano variety is finite.